Resonator

ABSTRACT

A micro-electro-mechanical device with a closed feed-back damping loop is provided. Displacement in the mechanical resonator is opposed with a damping force determined by the closed feed-back loop that comprises a filter with a peaked frequency response, and associated phase adjustment. An oscillation-free configuration that allows high signal amplification is achieved.

FIELD OF THE INVENTION

The present invention relates to mechanical resonators and especially toa control element, a mechanical resonator and a method for operating amechanical resonator.

BACKGROUND OF THE INVENTION

Micro-Electro-Mechanical Systems, or MEMS can be defined as miniaturizedmechanical and electro-mechanical systems where at least some elementshave a mechanical functionality. Since MEMS devices are created with thesame tools that are used to create integrated circuits, micromachinesand microelectronics can be fabricated on the same piece of silicon toenable machines with intelligence.

MEMS structures can be applied to quickly and accurately detect verysmall displacements, for example in inertial sensors. For example, in anaccelerometer a mass suspended on a spring structure to the body of thedevice may be displaced proportional to the acceleration of the device,and these displacements of the mass are detected. As a solid object, themass-spring structure typically has a resonant frequency in which itexhibits resonance or resonant behavior by naturally oscillating at somefrequencies, called its resonant frequencies, with greater amplitudethan at others. In these resonant frequencies the displacement is muchlarger than in other frequencies, which causes overload that disturbsthe detection in the miniaturized dimensions of MEMS structures.

These disturbances are typically eliminated by damping of the detectedmotion. A conventional method has been to use passive gas damping, butfor many applications gas damping is too non-linear and causes too manydisadvantageous effects to the operation of the system. In someconfigurations, like vibrating gyroscopes, gas damping is not evenapplicable, because damping to the resonant excitation of primaryvibration must be kept low.

In feed-back damping, or active damping, the detected displacement ismonitored and a relative force is generated to oppose the motion. Inknown systems, active damping has been implemented with a closedfeed-back loop that comprises a differentiator and a transducerresponsive to the differentiator signal. A differentiator has manyproperties due to which it is well suited to control damping ofdisplacements in mechanical resonators. The problem is, however, thatstructures are very seldom ideal, and in real-life resonators there areadditional mechanical resonance modes. When the differentiator outputsignal is amplified to generate an appropriately high damping force, thefeed-back-loop easily starts to oscillate disruptively.

BRIEF DESCRIPTION OF THE INVENTION

The object of the present invention is to provide an improved electricalforce feed-back mechanism for mechanical resonator systems. The objectsof the present invention are achieved with a control element, amechanical resonator and a method according to the characterizingportions of the independent claims.

The preferred embodiments of the invention are disclosed in thedependent claims.

The present invention is based on the idea of including in a dampingfeed-back loop a signal processing filter. The sign of the feed-back ofthe closed feed-back loop may be adjusted according to the type of thesignal processing filter. Preferably, the response function of thesignal processing filter has a resonant frequency characteristic, a peakthat essentially coincides with the resonant frequency of the dampedmechanical resonator, i.e. overlaps with the resonant frequency of themechanical resonator. A stable resonator that allows effectiveamplification of the detected signal is provided. The significantdifference in the resonating and non-resonating responses can be used toavoid unwanted oscillations in the closed feed-back loop of themechanical resonator.

BRIEF DESCRIPTION OF THE FIGURES

In the following the invention will be described in greater detail, inconnection with preferred embodiments, with reference to the attacheddrawings, in which

FIG. 1 illustrates a mass-spring system applicable for transducingdisplacements into electrical signals;

FIG. 2 illustrates another mass-spring system applicable for transducingdisplacements into electrical signals;

FIG. 3 shows a block diagram of a simplified exemplary sensing device;

FIG. 4 illustrates configuration of an exemplary mechanical resonator;

FIG. 5 shows a transfer function of an exemplary mechanical resonator;

FIG. 6 shows a phase transfer plot of an exemplary mechanical resonator;

FIG. 7 shows an exemplary conventional closed loop transfer function ofa damped mechanical resonator system;

FIG. 8 shows a phase transfer plot of a damped mechanical resonatorsystem;

FIG. 9 shows a schematic model of a basic 1-degree-of-freedom mechanicalresonator;

FIG. 10 shows a schematic model a mechanical resonator configurationwith additional mass-spring systems;

FIG. 11 shows a simulation printout of an exemplary transfer function;

FIG. 12 illustrates a device according to an embodiment of theinvention;

FIG. 13 shows transfer functions for an exemplary differentiator and anexemplary low-pass filter;

FIG. 14 shows amplitude response function for a closed loop with alow-pass filter;

FIG. 15 shows a phase response function for a closed loop with alow-pass filter;

FIG. 16 shows amplitude response function for a closed loop with ahigh-pass filter;

FIG. 17 shows a phase response function for a closed loop with ahigh-pass filter;

FIG. 18 shows amplitude response function for a closed loop with acombination of a band-pass filter and an all-pass filter; and

FIG. 19 shows a phase response function for a closed loop with acombination of a band-pass filter and an all-pass filter.

DETAILED DESCRIPTION OF SOME EMBODIMENTS

The following embodiments are exemplary. Although the specification mayrefer to “an”, “one”, or “some” embodiment(s), this does not necessarilymean that each such reference is to the same embodiment(s), or that thefeature only applies to a single embodiment. Single features ofdifferent embodiments may be combined to provide further embodiments.

In the following, features of the invention will be described with asimple example of a device architecture in which various embodiments ofthe invention may be implemented. Only elements relevant forillustrating the embodiments are described in detail. Variousimplementations of control elements, resonator devices and methodscomprise elements that are generally known to a person skilled in theart, and may not be specifically described herein.

A transducer refers here to a device that converts one form of energy toanother. For example, a sensing device or a sensor refers to anapparatus or a system that detects a physical property and converts itinto a measurable signal. Typically the physical property manifestsitself as one form of energy and is converted in the sensor to anotherform of energy with a transducer device. The physical property and themeans for detecting the form of energy may vary according to theapplication. In one category of sensors, a characteristic representing adetected physical property may comprise detection of displacement of areference point with respect of an inertial frame of reference. Thedisplacement may be detected, for example, as movement of a suspendedproof-mass, or as stress or strain exerted on a transducer elementconnected to an elastic suspension that carries the proof-mass. Thedetected displacement, stress or strain may be used to modulateelectrical signals, which makes the physical property quite accuratelymeasurable. As another example, in actuators, transducers may be used toconvert electrical energy into some form of motion.

A category of transducer devices applies a mass-spring system where themass is suspended on the spring to a body in such a manner that a forceexerted to the mass or acceleration of the body causes a proportionaldisplacement to the mass. By means of electric circuitry, movement ofthe mass may be detected and transformed into measurable electricalsignals. A mass-spring system is in an equilibrium state when the bodyis static and no net force acts on the mass. If the system is displacedfrom the equilibrium, the spring causes a net restoring force on themass, tending to bring it back to equilibrium. However, in moving backto the equilibrium position, the mass acquires a momentum which keeps itmoving beyond that position, establishing a new restoring force in theopposite sense. Hence, the mass begins to oscillate about theequilibrium state. The mass may be biased with a DC or AC voltage orcharge, and its movement in respect of static electrodes then convertedto an electrical signal by means of a voltage amplifier, chargeamplifier, transimpedance or transcapacitance amplifier, AC voltageamplifier, or the like. The harmonic oscillations of the mass thustransform into alternating electrical signals that represent movement ofthe body with respect to an inertial frame of reference.

FIG. 1 shows a simplified example of a mass-spring system applicable fortransducing displacements into electrical signals. The system may have aproofmass 10 and a spring element 11 anchored at anchor point 12 to asensed object (not shown). The system may have also one or morestationary electrodes 13 anchored to the body and one or more movableelectrodes 14 connected to the moving proofmass 10. The transducer formeasuring the displacement in FIG. 1 may be based on a capacitiveeffect, wherein capacitance between electrodes 13 and 14 changes whenthe distance between them changes. This change in capacitance may beused to modulate an electrical signal output from the transducer.

FIG. 2 shows another type of capacitive transducer; same referencenumerals are used to refer to corresponding elements in FIGS. 1 and 2.In the transducer of FIG. 2, modulation of the electrical signalcorresponds to change in the overlapping areas of the electrodes 13 and14. The transducers shown in FIGS. 1 and 2 measure displacement of aproof element based on a capacitive effect, but other types ofindicators, for example, strain or stress exerted on the proof element,and other types of effects, like piezoelectric, and electromagneticeffect that transform a detected indication of a physical property to anelectrical signal may be applied.

Solid objects typically have a natural frequency or group of frequenciesat which they tend to oscillate at greater amplitude than at others. Amass-spring system typically resonates and is therefore considered as aresonator. The resonance is a characteristic that may be useful for someapplications, but in other applications it may severely disturb theoperations. In inertial sensing, the mass-spring system easily overloadswhen the frequency of the vibration coincides with the resonantfrequency of the system. In order to avoid these unwanted side-effects,the resonance of the mass-spring system needs to be damped.

In some devices, gas damping is used to reduce effects of resonance.However, when striving for optimum performance with respect to noise andlinearity, gas damping is not the ideal mechanism, since gas damping mayhave a non-linear relationship to the displacement of the mass. Thenon-linearity may cause a translation of the vibration to an offsetshift, a phenomenon called vibration rectification. It may alsointroduce a new source of noise due to a specific dissipation mechanism.

Vibrating gyroscopes may also be modeled by mass-spring systems. Anyconcepts discussed herein for linear resonators can also be applied totorsional resonators, by replacing relevant masses with moments ofinertia, displacements with angles, and forces with torques. Theresonance effect is even more problematic with vibrating gyroscopes; gasdamping cannot be used because damping must be low for resonantexcitation of the primary vibration. In gyroscopes the problem has beenconventionally solved by increasing the Q-value (quality factor) of thesecondary resonator by thousands and dealing with the overload effectsby an overload margin.

Alternatively, a damping method comprising active electrical dampingwith a feed-back loop can be used. A displacement generates in the firsttransducer an electrical signal, which is converted in the feed-backloop into a modified signal that controls the mechanical force to beexerted against the displacement to eliminate or at least reduce thedetected movement. FIG. 3 shows a block diagram of a simplifiedexemplary resonator device 30 to illustrate how a feed-back loop ofactive electrical damping mechanism operates. The device of FIG. 3comprises a first mechanical resonator 31 that comprises a displacementsensing transducer TR1 and a forcing transducer TR2. FIG. 4 shows ablock diagram illustrating a more detailed configuration of theexemplary mechanical resonator 31 of FIG. 3.

The mechanical resonator of FIG. 4 comprises a proofmass 40, and aspring element 41 anchored at an anchor point 42 to a sensed object (notshown). The system has one or more stationary TR1 electrodes 43 anchoredto the body and one or more movable TR1 electrodes 44 connected to themotion of the proofmass 40. These electrodes 43, 44 form a capacitivedisplacement measuring transducer, as will be explained with FIG. 3. Thesystem has also one or more stationary TR2 electrodes 45 anchored to thebody and one or more movable TR2 electrodes 46 connected to the motionof the proofmass 40. These electrodes 43, 44 form a force generatingelectrostatic transducer.

Capacitive and electrostatic transducers in FIG. 4 are only examples.The transducers TR1 and TR2 of FIG. 3 may be based on any transducerprinciple. For example, the first transducer may be configured to detectdisplacements by means of motion, stress or strain based on capacitiveeffect, piezoelectric effect, electromagnetic effect, or piezoresistiveeffect. The second transducer may be configured to induce displacementsby means of force, torque, stress or strain based on electrostaticeffect, piezoelectric effect, electromagnetic effect, thermoelasticeffect, electrostriction, or magnetostriction.

It is known to a person skilled in the art that the same physicaltransducer may operate either as a displacement to electrical signalconverter or an electrical signal to force converter, depending on thecircuits attached to the transducer. It is also known that one singletransducer can operate in both roles simultaneously using e.g. timedivision (multiplexing) or frequency division technique.

As discussed earlier, in prior art solutions, a wide band phase shiftingelement (a differentiator) has been used in the feed-back loop. It hasnow been detected that the desired damping effect can be provided evenwith feed-back that is only on a narrow frequency band around theresonant frequency. Accordingly, the following embodiments describe howto limit the bandwidth of the feed-back control element and still obtainthe desired damping effect. This is achieved by introducing a controlelement with a peaked frequency response, i.e. a response where theamplification is high over a narrow frequency band, hereinafter called apass band, and much lower above and below the pass band. The frequencyof the resonance that is to be damped is adjusted to fall within thispass band, but exact coincidence with the peak frequency or with thecenter of the pass band is not required.

There are many ways to accomplish the desired peaked narrow bandfrequency response of the control element by means of continuous time,discrete time, tapped delay line and other such known filter concepts inanalog or digital domain. Not all filter characteristics are suitablefor a narrow band feed-back loop. The phase shift of the controlelement's frequency response within the peaked frequency band must notbe too large with respect to the phase at the resonant frequency inorder not to cause instability within the frequency band. Only simplelow order characteristics are thus applicable.

Returning to FIG. 3, a displacement of a proofmass in the mechanicalresonator 31 may be measured electrically with TR1 electrodes (43,44 inFIG. 4) and converted to an electrical signal S1. The first electricalsignal S1 may be amplified by an amplifier 32, and the amplified signalfed via a controller 33 to the forcing transducer TR2. TR2 is configuredto exert with TR2 electrodes (45,46 in FIG. 4) on the proofmass amechanical force that corresponds to the second electrical signal S2. S2is fed to TR2 having such a phase relation to the detected displacementthat movement of the proofmass is damped, i.e. reduced by the appliedforce.

The transfer function of the mechanical resonator 31 of FIG. 3 is:

$\begin{matrix}{{H_{m}(s)} = \frac{1}{1 + {s/Q_{m}} + s^{2}}} & (1)\end{matrix}$

where Q_(m) is the mechanical Q-value of the mechanical resonator.Frequency has been normalized so that the resonant frequency isrepresented by ω₀=2πf₀=1, and s represents the normalized frequency(imaginary number). FIG. 5 shows a transfer function of an exemplarymechanical resonator with Q-value 1000, and FIG. 6 shows a phasetransfer plot of the same. It is seen that when the mechanical resonatoris excited at its resonant frequency, the displacement reached with aspecific driving force is maximized. FIGS. 5 and 6 show that there is avery narrow and high peak at the resonant frequency in the amplituderesponse and a steep transition from 0 to −π in the phase response.

Conventionally higher damping can be achieved by increasing theamplification in the feed-back loop, i.e. so called loop gain thatincludes contributions from any amplifiers in the loop, the controller,the transducers, the frequency response of the mechanical resonator, andthe spring constant of the resonator spring element. However, whiledoing so, one has to make sure that the closed loop transfer functionremains stable at all conditions. This may be ensured by selecting aproper transfer function to the controller. A conventional transferfunction for this is the differentiator:

H _(C)(s)=s  (2)

A differentiator is basically an ideal controller since it produces aconstant +π/2 phase shift at all frequencies. This means that incombination with a transfer function of equation (1) good phase marginfor closed loop operation may be ensured when negative feed-back isapplied. The closed loop transfer functions of (1) and (2) together inthe system of FIG. 3 reduce the mechanical Q-value of the system todesired levels, which allows the system to be stable:

$\begin{matrix}{{H(s)} = \frac{Ks}{1 + \left( {K + {1/Q_{m}}} \right) + s + s^{2\;}}} & (3)\end{matrix}$

where K is the loop gain at low frequency. The effective Q-value of theclosed loop is

$\begin{matrix}{Q_{eff} = {\frac{1}{K + {1/Q_{m}}} \approx \frac{1}{K}}} & (4)\end{matrix}$

FIG. 7 shows a closed loop transfer function of the damped mechanicalresonator system 30 shown in FIG. 3 in a case where the control element33 is a differentiator. FIG. 8 shows a phase transfer plot of the same.The mechanical Q-value in FIGS. 7 and 8 has been set to 1000, the lowfrequency amplification is 1, and the controller CTRL has a transferfunction of equation (2).

However, when the controller has a transfer function of equation (2),and when higher values of amplification are attempted, practical systemseasily start to oscillate at a frequency that is considerably higherthan the resonant frequency of the resonator. This oscillation is causedby additional mechanical resonance modes due to real-life resonators.These modes may be caused e.g. by the flexibility of the mass, byresonances caused by the transducer, deflection in harmonic modes of thesprings, deflection in torsion modes of the springs, and similar othernon-idealities. Such non-idealities typically cause additional resonancepeaks to be created in the transfer function.

FIG. 9 shows a schematic model of a basic 1-degree-of-freedom mechanicalresonator formed of a proofmass 90, a massless spring 91, and a dashpotdamper 92. The above mentioned non-idealities can be simulated byincluding to the configuration additional mass-spring systems 100, 101as shown in FIG. 10 (for simplicity, the possible mechanical damping isexcluded in FIG. 10). FIG. 11 shows an exemplary transfer functionreceived as a result of simulations with a configuration that includesadditional masses to represent these real-life non-idealities. FIG. 11shows typical additional resonant frequencies where oscillation startswhen there is enough loop gain and when the phase shift is such that theonset of an oscillation is promoted.

One of the reasons for the observed on-set of oscillations at otherresonance modes is that a differentiator typically emphasizes highfrequencies. Due to this emphasis, only very moderate levels ofamplification have conventionally been possible, and thereby adequatedamping is not yet achieved. There has been attempts to compensate forthis property of the differentiator by filtering the higher frequenciesin the controller (e.g. Toshiki Hirano in Jpn. J. Appl. Phys. Vol. 42(2003) pp. 1486-1490 Part 1, No. 3, March 2003), but the results are notadequate for many practical applications where the additional resonantfrequencies may be close to the main resonant frequency.

FIG. 12 illustrates a device configuration and at the same time steps ofa method for controlling operation of a mechanical resonator accordingto an embodiment of the invention. The configuration is similar to theconfiguration of FIG. 3, but here feed-back force against the detectedmotion is controlled with a signal processing filter 123.Advantageously, the signal processing filter 123 functions as aresonator, i.e. a resonating filter, and therefore peaks at a definedresonant frequency. Accordingly, the response function that definescorrespondence between values of S1 and S2 is a frequency responsefunction that has a resonant frequency characteristic that essentiallycoincides with the resonant frequency of S1.

In signal processing, a filter refers to a device or a process thatcompletely or partially suppresses from a signal some unwanted componentor feature. A signal processing filter (“filter”) with frequencyresponse is configured to remove from a signal some frequencies and notothers. The transfer function of the filter is a frequency responsefunction that defines a relation between a signal that it inputs and asignal that it outputs. A cutoff frequency of the filter is a frequencyafter which the filter begins to attenuate or eliminate signals.Roll-off of the filter defines steepness of the response function withfrequency after the cut-off frequency. It is known that roll-off tendstowards a constant gradient at frequencies well away from the cut-offfrequency. Roll-off can occur with decreasing frequency as well asincreasing frequency, depending on the type of the filter. For example,a low-pass filter will roll-off with increasing frequency, but ahigh-pass filter will roll-off with decreasing frequency.

In the present embodiment, the signal processing filter is a resonator,i.e. a resonating filter with a specific resonant frequency just beforeits cut-off frequency. As shown in FIG. 13 graph 131, this means thatthe frequency response is high at the resonant frequency of the filterand steeply rolls-off with increasing frequency. In regions in the otherside of the cut-off frequency (later: low-frequency response region),the response function is more even, but the difference to the resonanceresponse is still considerable.

It has been discovered that the significant difference in the responseof the filter to frequencies in and closely around the resonantfrequency and to frequencies beyond them can be applied to avoidunwanted oscillations in the closed feed-back loop of the mechanicalresonator. A significantly higher level of damping may be achieved byutilizing this difference. The principle is illustrated in FIG. 13 thatshows transfer functions for an exemplary differentiator 130 and for anexemplary low-pass filter 131. It may be seen that in the high frequencyrange above the resonant frequency, where the additional resonantfrequencies typically are, the difference between the responses isconsiderable, in the order or 100 or more. Accordingly, in this highfrequency range the differentiator emphasizes signals, but theresonating filter very effectively attenuates them. Oscillations causedby additional resonant frequencies in this high frequency range maytherefore be effectively eliminated with a resonating filter.

It is also seen that attenuation of the mechanical resonator in thelow-frequency response range may not be needed since the main resonanceis typically the lowest resonant frequency of the device. In order toeliminate adverse effects from unwanted phase shift in that region,amplification of the signal by the resonator has to be controlled.Returning back to FIG. 12, the block chart shows a first resonator R1121 that, as in FIG. 3, is a mechanical resonator that produces a firstelectrical signal S1. The frequency of S1 corresponds to vibration of aproofmass in respect of a frame of reference. S1 may be pre-amplifiednormally by a first amplifier 122, and the amplified signal fed to asecond resonator 123 R2. The second resonator R2 generates, according toits frequency response function, a modified electrical signal S2. S2 maybe amplified by the second amplifier and then fed to R1 to define theamount of damping force to be exerted on R1. In some conventionalconfigurations, the primary mechanical system has been stimulated by oneor more forcers at the resonant frequency of one of its modes to make itoscillate with constant amplitude. In the present invention, however,resonant frequency response is applied to generate a feed-back dampingforce that opposes the detected motion.

In order to ensure that the damping occurs at an appropriate phase, thesign of the feed-back may be adjusted according to the type of filter.Let us consider first a case where controller 123 is a low-pass filter.The phase shift of the mechanical resonator at the resonant frequency is−π/2 and the phase shift of the low-pass filter is −π/2. For stableoperation, the phase shift of the feed-back loop may be −π. This meansthat in the case of a low-pass filter, the sign of the feed-back may bepositive at low frequencies. The sign of the feed-back-loop may bereversed with mechanisms well known to a person skilled in the art, forexample with amplification stages, or appropriate adjustment of otherloop parameters such that the signals are summed instead of subtracted.In the roll-off region, the fast decrease in response levels ensuresoscillation-free operation. It is, however, well known that positivefeed-back may cause instability in the low-frequency response region,which in certain embodiments of the invention would entail the low-passfilter in the low frequency range.

Consequently, in order to avoid instability by signals in the lowfrequency range, their amplification may be set to a low level,preferably to values less than 1. By selecting the electrical Q-value ofthe resonating low-pass filter from a range of 3 to 10, and arrangingthe amplitude peak of the low-pass filter to essentially coincide withthe resonant frequency of the mechanical resonator R1, the amplificationof R2 at low frequency can be reduced well below 1, and still the loopgain around the resonant frequency can be increased with AMP2 to a highenough level for efficient damping.

An advantageous low-pass transfer function for the controller is asecond order function and may have the form:

$\begin{matrix}{{H_{LP}(s)} = \frac{K}{1 + {s/Q_{e}} + s^{2}}} & (5)\end{matrix}$

where Q_(e) is the Q-value of the resonating low-pass filter and K isthe amplification at low frequency. Q_(e) is preferably in the rangefrom 3 to 10 and K is in the range from 0.1 to 0.3. Equation (5)corresponds to a second order continuous time filter that can beimplemented either by analog or digital means. It is well known to aperson skilled in the art how to implement essentially similar frequencycharacteristics by a discontinuous time filter, tapped delay line filteror other known filter types. When the transfer function of the equation(5) is used in a feed-back system and the sign of the amplification inthe feed-back loop is selected as positive, one obtains a closed looptransfer function as shown in FIG. 14 for the amplitude and in FIG. 15for the phase.

It may be seen that the amplitude response has two peaks around theresonant frequency of the mechanical resonator, but they are not so highas to compromise the damping of the original very high Q-valueresonance. As a further advantage, the matching of the peak of thecontroller response to the mechanical resonance does not have to be veryprecise. A 10% deviation does not cause a noticeable effect, and adeviation as high as 30% may still be useful. Essential coincidence mayin this context be interpreted to mean that the resonant frequency inwhich the transfer function of the second resonator (controller R2)reaches its highest value may deviate to some extent from the resonantfrequency of the mechanical resonator R1. Advantageously, the deviationremains under 20% such that the resonant frequency of R2 remains in therange of 80% to 120% of the resonant frequency of R1. However, even a50% deviation, i.e. R2 values in the range of 50% to 150% of theresonant frequency of R1 are applicable for some applications ofdamping.

Due to the tendency of the additional resonant frequencies to appear inhigher frequencies, the low-pass filter operates very well in thefeed-back loop because the fast roll-off efficiently eliminates theunwanted elements from the signal. By adjusting the amplification of thecontroller low enough, it is also possible to eliminate the instabilityin the low frequency range. As such, a damping signal that provides thecorrect phase shift and can be strongly amplified is achieved.

The same concept may be applied to other types of filters, as well. Inanother embodiment, controller 33 may be implemented with a high-passfilter. In this case, the phase shift in the feed-back loop from themechanical resonator is −π/2, and the phase shift of the high-passfilter is +π/2. This means that in the case of a high-pass filter, otheraspects of the closed loop may be designed in the similar manner as inthe low-pass filter case, but the sign of the amplification in thefeed-back loop may now be set to be negative. A transfer function forthe second order high-pass configuration may have the form:

$\begin{matrix}{{H_{HP}(s)} = \frac{{Ks}^{2}}{1 + {s/Q_{e}} + s^{2}}} & (6)\end{matrix}$

With negative feed-back the resulting amplitude and phase transferfunctions become as shown in FIGS. 16 and 17 respectively. Equation (6)corresponds to a continuous time filter that can be implemented eitherby analog or digital means. It is known to a person skilled in the arthow to implement essentially similar frequency characteristics by adiscontinuous time filter, tapped delay line filter or other knownfilter types.

In another embodiment, controller 33 may be implemented with acombination of a band-pass filter and an all-pass filter. In this case,the phase shift in the feed-back loop from the mechanical resonator is−π/2, and the phase shift of the band-pass filter is 0. The band-passfilter provides appropriate frequency response characteristics, butappropriate phase shift values cannot be provided. The phase shift of anall-pass filter is −π/2 with no effect on roll-off, so with an addedall-pass filter, the desired frequency response characteristics can bemaintained, and the correct phase shift provided when the sign of theamplification of the feed-back loop is set positive. Other aspects ofthe closed loop may be designed in a similar manner to the low-passfilter. A transfer function for the second order band-pass/all-passconfiguration may have form

$\begin{matrix}{{H_{{BP}\text{-}{AP}}(s)} = {\frac{Ks}{1 + {s/Q_{e}} + s^{2}} \cdot \frac{1 - s}{1 + s}}} & (7)\end{matrix}$

When positive feed-back is applied the resulting amplitude and phasetransfer functions are as shown in FIGS. 18 and 19 respectively.Equation (7) corresponds to a continuous time filter that can beimplemented either by analog or digital means. It is known to a personskilled in the art how to implement essentially similar frequencycharacteristics by a discontinuous time filter, tapped delay line filteror other known filter types.

It is apparent to a person skilled in the art that as technologyadvances, the basic idea of the invention can be implemented in variousways. The resonator device may be a sensor device, like anaccelerometer, an angular rate sensor, a magnetic field sensor, anactuator device, like an opto-mechanical device, or a switching device.Furthermore, for simplicity, only lowest order filters with suitablefrequency responses have been discussed herein. Higher order filterswith appropriate resonance and phase characteristics are well within thescope of this invention.

The disclosed examples have been illustrated with continuous time filterwith a second order transfer function that can be implemented either byanalog or digital means. As well known to a person skilled in the art,it is possible to implement essentially similar frequencycharacteristics with a discontinuous time filter, tapped delay linefilter or other known filter types. It is also possible to use higherthan second order filters e.g. by placing additional high-pass and/orlow pass filters in series with a second order resonating filter. Anadditional high pass filter with the low frequency cut-off well belowthe peak frequency of the resonating filter may be advantageous foreliminating the need for DC amplification, and thus avoiding possibleoperating point drift of the amplifying circuits. The resonating filtermay also have higher than second order transfer function if in itsdesign care is taken to avoid instability within the pass band. Part ofthe closed loop frequency characteristics can also be implemented in asignal amplifier instead of the feed-back controller.

The invention and its embodiments are therefore not restricted to theabove examples, but may vary within the scope of the claims.

1. A micro-electro-mechanical device, comprising: a resonator; afeed-back loop; wherein the feed-back loop is adjusted to damp theresonator with an electrical force feed-back mechanism; and damping bythe feed-back loop is controlled by a filter with a peaked frequencyresponse that peaks at a defined resonant frequency.
 2. Amicro-electro-mechanical device according to claim 1, whereinamplification of the peaked frequency response in a pass band offrequencies is multifold to amplification of the peaked frequencyresponse beyond the pass band of frequencies.
 3. Amicro-electro-mechanical device according to 2, wherein a mechanicalresonant frequency of the resonator is within the bass band of thepeaked frequency response.
 4. A micro-electro-mechanical deviceaccording to claim 1, wherein the quality factor of the filter isabove
 1. 5. A micro-electro-mechanical device according to claim 1,wherein the quality factor of the filter is in the range 3 to
 10. 6. Amicro-electro-mechanical device according to claim 1, wherein a loopgain of the feed-back loop for frequencies below the resonant frequencyis less than
 1. 7. A micro-electro-mechanical device according to claim1, wherein a loop gain of the feed-back loop for frequencies below theresonant frequency is in the range 0.1 to 0.3.
 8. Amicro-electro-mechanical device according to claim 1, wherein the filteris a low-pass filter and the sign of the feed-back amplification of thefeed-back loop is positive.
 9. A micro-electro-mechanical deviceaccording to claim 1, wherein the filter is a high-pass filter and thesign of the feed-back amplification of the feed-back loop is negative.10. A micro-electro-mechanical device according to claim 1, wherein thefilter is a combination of a band-pass filter, and an all-pass filterand the sign of the of the feed-back amplification of the feed-back loopis positive.
 11. A micro-electro-mechanical device according to claim 1,wherein the micro-electro-mechanical device is a sensing device.
 12. Amicro-electro-mechanical device according to claim 1, wherein themicro-electro-mechanical device is a sensor of angular motion.
 13. Amicro-electro-mechanical device according to claim 1, wherein themicro-electro-mechanical device is a sensor of linear motion.
 14. Amicro-electro-mechanical device according to claim 12, wherein themicro-electro-mechanical device is configured to detect displacement bymeans of motion, stress or strain based on capacitive effect,piezoelectric effect, electromagnetic effect or piezoresistive effect.15. A micro-electro-mechanical device according to claim 1, wherein thefeed-back loop is configured to induce damping by means of force,torque, stress or strain based on electrostatic effect, piezoelectriceffect, electromagnetic effect, thermoelastic effect, electrostrictionor magnetostriction.
 16. A method for operating amicro-electro-mechanical device, comprising a resonator and a feed-backloop; wherein the method comprises damping the resonator with anelectrical force feed-back mechanism of the feed-back loop; andcontrolling damping by the feed-back loop by a filter with a peakedfrequency response that peaks at a defined resonant frequency.
 17. Amethod according to claim 16, further comprising adjusting a mechanicalresonant frequency of the resonator within a bass band of the peakedfrequency response, wherein amplification of the peaked frequencyresponse in the pass band of frequencies is multifold to amplificationof the peaked frequency response beyond the pass band of frequencies.18. A micro-electro-mechanical device according to claim 13, wherein themicro-electro-mechanical device is configured to detect displacement bymeans of motion, stress or strain based on capacitive effect,piezoelectric effect, electromagnetic effect or piezoresistive effect.